Insofar as balrog was a word Tolkien invented, the name (used for the two boss characters in Street Fighter II: The World Warrior) almost certainly goes back ultimately to Tolkien.* However, as a generic name for a powerful monster, it had already passed into common use by the 1980s. Street Fighter II was at least the fourth video game I had played that used "balrog" as a monster name. The most powerful enemy in the original Ultima was a balron (changed, along with other explicit Tolkien references from "balrog" in Lord British's previous game Akalabeth: World of Doom).

Garriot's balrogs were very clearly influenced by Tolkien, as was the frequently-renamed balrog, type VI demon, balor, roaring demon that appeared in Dungeons & Dragons. See how it carries the same "whip of many thongs" as Durin's Bane.

However, well before Street Fighter II, there were also instances where the name had moved far from its roots in The Lord of the Rings. There was this as-yet-unidentified game, that was probably a homebrew creation of someone associated with the Oregon Museum of Science and Industry. The goal there was to kill the balrog. Balrogs also appeared in (fixed, but not random) encounters in the hardest dungeon of The Bard's Tale.

So, whether or not some or all of the game creators at Capcom were aware that balrog was Tolkien's coinage, the name was already "out there" in the tabletop and video game communities—with the connection to The Lord of the Rings sometimes completely bleached away.

*The Oxford English dictionary has no entry for balrog, but it does appear in a 1994 quote from The Nation—although in a context that suggests that what is being described is a human (or humanoid) video game character:

On the video screen, a plug-ugly named Balrog had just cut loose with a series of side-kicks and elbow blows.

That sentence could even be a reference to Street Fighter II itself!

Insofar as balrog was a word Tolkien invented, the name (used for the two boss characters in Street Fighter II: The World Warrior) almost certainly goes back ultimately to Tolkien.* However, as a generic name for a powerful monster, it had already passed into common use by the 1980s. Street Fighter II was at least the fourth video game I had played that used "balrog" as a monster name. The most powerful enemy in the original Ultima was a balron (changed, along with other explicit Tolkien references from "balrog" in Lord British's previous game Akalabeth: World of Doom).

Garriot's balrogs were very clearly influenced by Tolkien, as was the frequently-renamed balrog, type VI demon, balor, roaring demon that appeared in Dungeons & Dragons. See how it carries the same "whip of many thongs" as Durin's Bane.

Another example, pointed out by MartianInvader in the comments, was the Chaos gargoyle figurine from the Hero Quest game, which is a winged bipedal monster wielding a whip and a jagged sword.

However, well before Street Fighter II, there were also instances where the name had moved far from its roots in The Lord of the Rings. There was this as-yet-unidentified game, that was probably a homebrew creation of someone associated with the Oregon Museum of Science and Industry. The goal there was to kill the balrog. Balrogs also appeared in (fixed, but not random) encounters in the hardest dungeon of The Bard's Tale.

So, whether or not some or all of the game creators at Capcom were aware that balrog was Tolkien's coinage, the name was already "out there" in the tabletop and video game communities—with the connection to The Lord of the Rings sometimes completely bleached away.

*The Oxford English Dictionary has no entry for balrog. (If it were an Old English word that Tolkien had resurrected, the OED ought to include it. Their lexicographers are not perfect, but they seem pretty good about words used by Tolkien.) However, the name does appear once in the OED's citations—in a 1994 quote from The Nation about the boxer character from Street Fighter II itself!

On the video screen, a plug-ugly named Balrog had just cut loose with a series of side-kicks and elbow blows.